UM ETheses Collection (澳門大學電子學位論文庫)
 Title

PFST(MT) 000 (SAMPLE) Center of mass distribution of the Jacobi unitary ensembles, asymptotic expansions and Heun equations
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Abstract We investigate the probability density function, P(c, α, β, n) dc, of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑nj=1 f (xj ) := ∑nj=1 xj , denoted by Mf (λ, α, β, n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β , x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the timeevolved Jacobi weight, namely, w(x; λ, α, β) = xα(1 − x)β e−λ x, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The nonclassical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + · · · + Pn(0, λ), orthogonal with respect to w(x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor, Chen and Ehrhardt [6], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w(x; λ, α, β)/x. It is shown that rn(2iez ) satisfies a Chazy II equation. There is another auxiliary variable, denote as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w(x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: Pv(α²/2, −β²/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ−form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function Mf (λ, α, β, n) when n is sufficiently large. Furthermore, we deduce a new expression of Mf (λ, α, β, n) when n is finite in terms the σ function of this the Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the PaleyWiener theorem, one deduces that P(c, α, β, n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as w the weight is positive and continuous over [0, 1]. Secondly, we studied a system of second order different equation, the coefficients are rational functions, satisfied by orthogonal polynomial Pn(z) associated with the related weights. Jacobi type: 1)xα(1 − x)β e−tx, x ∈ [0, 1], α, β, t > 0; 2)xα(1 − x)β e−t/x, x ∈[0, 1], α, β > 0, t ≥ 0; 3)(1−x²)α(1−k²x²)β , x ∈ [−1, 1], α > −1, k² ∈ (0, 1); Laguerre type: 1)xα(x + t)λe−x, x ∈ [0, ∞), t, α, λ > 0; 2)xαe−x−t/x, x ∈ [0, ∞), α, t > 0; Weights with gap: 1)e−x²(1 − χ(−a,a)(x)), x ∈ R, a > 0; 2)(A + Bθ(x −t))xαe−x, x ∈ [0, ∞), t > 0, A ≥ 0, A+B ≥ 0; 3)(1−x²)α(1−χ(−a,a)(x)), x ∈[−1, 1], a ∈ (0, 1), α > 0. We consider n, the dimension of the Hankel matrix tends to ∞ (infinite dimension). The second order differential equations satisfied by Pn(z), after asymptotic, are Heun equations. That means the orthogonal polynomial is the solution of the Heun equation. Heun equation may write as Hamiltonian structure, then from the Hamiltonian structure deduce the Painleve equation, namely, After asymptotic the coefficients of second differential equation may write as Heun equation and Painleve equations. As we know the coefficients, αn(t) and βn(t), of three term recurrence relation satisfy Painleve equation. Obvious these two type of Painleve equations are not the same, as the behaviours of asymptotic the recurrent coefficients represent as Taylor serious and the Painleve equation in terms with orthogonal polynomials and the latter one in terms with the recurrence coefficients. The weight we mentioned in our paper were studied by Y. Chen and his collaborators [7, 8, 16, 17, 52, 53, 91]. We choice some classic cases to show the second order differential equation in terms with Pn(z) were Heun class equations, also could written as Painleve class equations, after asymptotic. Heun equation is of considerable importance in mathematical physics, since the special case of Heun’s equation include the Gauss and confluent hypergeometric, Mathieu, Ince, WhittakerHill, Lame, Bessel, Legendre, Lagureee equations, [1, 43, 73] etc. Further more, we investigate the discrete semiclassical orthogonal polynomials of class s = 1, associated with the weight function w(x) = (α1)x(α2)x(α3)x(β1)x(β2)x1 x!, αi, βj > 0, i = 1, 2, 3, j = 1, 2 also can denoted by Pn3,2 (x; α1, α2, α3, β1, β2; 1). We show the difference equation satisfied by the recurrence coefficients of discrete orthogonal polynomials.
 Issue date

2019.
 Author

Zhan, Long Jun
 Faculty

Faculty of Science and Technology
 Department

Department of Mathematics
 Degree

Ph.D.
 Subject
 Supervisor

Chen, Yang
 Location
 1/F Zone C
 Library URL
 991008148679706306